It’s not often that one can report mathematics news but two weeks ago a new Mersenne prime was discovered. A Mersenne prime is a prime number of the form ie one less than a power of two. It is easy to show that itself must also be prime. The new prime number is has 7,235,733 decimal digits and was found on a 2.4 GHz Pentium 4 computer running Windows XP.

At A level students only see reasonably nice functions, though occasionally they come across

where the maximum is smaller than the minimum (which, incidentally, is why one should use the terms local maximum and local minimum instead).

But what they won’t have come across are functions like these:

When trying to draw the graphs remember that

between every pair of irrational numbers there is a rational number

between every pair of rational numbers there is a irrational number

You can very roughly draw f using dots and there’s a nice picture of g at mathworld
Now think about integrating f and g – do they have areas under between them and the x-axis? The answer to ths question is important in the theory of integration.

These functions have other interesting properties and illustrate the relationship between rational and irrational numbers.

You have to connect the dots by using 4 lines, without taking your pen off the paper and only going through each dot once. At first sight this looks impossible until you are shown you can go outside the square formed by the dots as in

But what about 16 dots arranged in a square, or 12 dots arranged in 3 rows of 4? How many lines are needed for a 10 x 10 grid? What if arcs of circles are used instead of lines?

These and other fascinating questions about dots can be found at the web page simply called dots